Can models get 100% accuracy on solved games?
So can we perfectly learn a solved game as well?

The short answer is yes. If your model has enough complexity it can theoretically learn any behavior you want.

So in theory our model could learn tic-tac-toe

Tic Tac Toe has already been solved. Another popular game that has been solved is Checkers, by the algorithm Chinook.
To be more specific, in Reinforcement Learning we make the assumption that any decision making process can be modeled as an MDP (Markov Decision Process). Once there, there are a host of different methods like Q-Learning and TD that theoretically converge towards the optimal policy - the one the plays perfectly.
Now, just because it is theoretically possible doesn't mean it will always empirically work. In games that are very complex and have a large state space it is extremely difficult to perfectly solve. This is because the only feasible way to go about them is to approximate and getting perfect play even in small edge cases becomes much more difficult as a result.
If you want to learn more about this topic I would highly recommend this series RL Course by David Silver